Modal Analysis of a Spinning Disk in a Dense Fluid as a Model for High Head Hydraulic Turbines
MAX LOUYOT
Département de génie mécanique
Mémoire présenté en vue de l’obtention du diplôme de Maîtrise ès sciences appliquées Génie aérospatial
Août 2019
c
Ce mémoire intitulé :
Modal Analysis of a Spinning Disk in a Dense Fluid as a Model for High Head Hydraulic Turbines
présenté par Max LOUYOT
en vue de l’obtention du diplôme de Maîtrise ès sciences appliquées a été dûment accepté par le jury d’examen constitué de :
Jean-Yves TRÉPANIER, président
Frédérick P. GOSSELIN, membre et directeur de recherche Stéphane ÉTIENNE, membre
DEDICATION
ACKNOWLEDGEMENTS
Firstly, I would like to thank my research director Frédérick P. Gosselin for his continuous interest in my work and his help through these couple years in spite of an entire continent separating us. I would also like to thank my office mate Alexandre Couture for his smart suggestions, an always pleasant company, and the pastries of course.
Another warm thank to Bernd Nennemann for his quality supervision at Andritz Hydro and for helping me feel comfortable in the CFD group; group that I would also like to thank for their help in many different aspects of my work: Samer Afara, Maxime Gauthier, Matthieu Melot and Marine Heschung. For the same reason I would like to thank Melissa Fortin, Christine Monette and our european collegues Beat Horisberger and Olivier Braun.
My sincere acknowledgements to the MITACS Accelerate program and to Andritz Hydro for financing my research during this Master’s. Lastly, I wish to thank École Polytechnique de Montréal and Université de Montréal for giving me the opportunity to go through this rich experience in the most pleasant context as a student.
RÉSUMÉ
Dans les turbines Francis de hautes chutes et dans les pompes-turbines en particulier, les Interactions Rotor Stator (RSI) sont une source d’excitation inévitable qui doit être pré-dite avec précision. Une connaissance pointue des caractéristiques dynamiques des turbines, notamment de la variation des fréquences naturelles du rotor en fonction de la vitesse de ro-tation et de la masse ajoutée de l’eau avoisinante, est essentielle à la prédiction de résonances potentielles et de l’amplification des vibrations résultantes. Dans ces machines, la couronne et la ceinture de la roue ainsi que les flasques supérieur et inférieur possèdent une structure similaire à un disque, ce qui donne lieu à l’apparition de modes diamétraux et à un phéno-mène de séparation des fréquences pour lequel aucune méthode de prédiction efficace n’existe à ce jour. Les méthodes d’Interactions Fluide-Structure (IFS) complètement couplées coûtant trop cher en temps de calcul, un modèle simplifié basé sur l’approche de force modale serait un outil puissant en terme de design et de prédiction de temps de vie de ces turbines. Ce travail présente le développement d’un modèle analytique et d’un modèle de Mécanique des Fluides Numérique (CFD) pour les disques en rotation dans un fluide dense, permettant la prédiction précise de la séparation et du décalage des fréquences qui sont observés expé-rimentalement. De plus, une explication sur l’origine physique du phénomène de séparation des fréquences en est déduite. Ces modèles sont validés par comparaison avec des données expérimentales.
ABSTRACT
In high head Francis turbines and pump-turbines in particular, Rotor Stator Interactions (RSI) are an unavoidable source of excitation that needs to be predicted accurately. Precise knowledge of turbine dynamic characteristics, notably the variation of the rotor natural frequencies with rotation speed and added mass of the surrounding water, is essential to assess potential resonance and resulting amplification of vibrations. In these machines, the disk-like structures of the runner crown and band as well as the head cover and bottom ring give rise to the emergence of diametrical modes and a mode split phenomenon for which no efficient prediction method exists to date. Fully coupled Fluid-Structure Interaction (FSI) methods are too computationally expensive; hence, a simplified method based on the modal force approach would be a powerful tool for the design and expected life prediction of these turbines.
This work presents the development of both an analytical and a numerical Computational Fluid Dynamics (CFD) model for a rotating disk in dense fluid, which accurately predict the natural frequency split as well as the natural frequency drift that are observed empirically. Additionally, insight is given on the physical origin of the mode split phenomenon. These models are validated by comparison with experimental data.
TABLE OF CONTENTS
DEDICATION . . . iii
ACKNOWLEDGEMENTS . . . iv
RÉSUMÉ . . . v
ABSTRACT . . . vi
TABLE OF CONTENTS . . . vii
LIST OF TABLES . . . viii
LIST OF FIGURES . . . ix
LIST OF SYMBOLS AND ACRONYMS . . . xii
LIST OF APPENDICES . . . xiii
CHAPTER 1 INTRODUCTION . . . 1
CHAPTER 2 METHODOLOGY . . . 6
2.1 Structural model . . . 6
2.2 Fluid-Structure Interactions (FSI) analytical model . . . 9
2.3 FSI numerical model . . . 14
CHAPTER 3 RESULTS AND DISCUSSION . . . 19
3.1 Analytical model results . . . 19
3.2 Numerical model results . . . 24
CHAPTER 4 CONCLUSION . . . 29 4.1 Summary of Works . . . 29 4.2 Limitations . . . 30 4.3 Future Research . . . 30 REFERENCES . . . 31 APPENDICES . . . 36
LIST OF TABLES
Table 3.1 Parameter values for modeling the Presas et al. (2015) experimental test rig geometry. The disk is made of stainless steel; the fluid is water. 19 Table 3.2 Natural frequencies of modes n = ± 2, 3, 4, s = 0 obtained with the
analytical model Eq. (2.52) and the Presas et al. (2015) experiments for different disk rotation speeds and the corresponding test rig geometry, and relative error . f = |ω|/2π. . . . 19 Table 3.3 Natural frequencies of modes n = 2, 3, 4, s = 0 obtained with the
analytical model Eq. (2.52), the numerical model Eq. (2.19), and ex-periments from Presas et al. (2015) for the stationary disk in water, and relative error . f = |ω|/2π. . . . 27 Table 3.4 Split magnitude of modes n = ± 2, 3, 4, s = 0 obtained with the
an-alytical model Eq. (3.1) and the numerical model Eqs. (2.56-2.57) for the Presas et al. (2015) rotating disk at ΩD/2π = 40 Hz in water, and
LIST OF FIGURES
Figure 1.1 (a) A turbine runner geometry. (b) High head Francis and pump-turbine runners have disk-like modeshapes, characterized by their num-ber n of nodal diameters (n = 3 is presented). (c) Each disk mode is composed of a co-rotating and a counter-rotating wave, with respect to the fluid rotation relative to the disk. (d) Co- and counter-rotating wave frequencies evolve with the disk rotation speed in dense fluid: the split between the two increases, while the average value decreases. . . 3 Figure 2.1 Studied geometry in §2.1 and §2.2: the disk with angular speed ΩD,
outer radius a, inner radius b and thickness h is confined in a rigid casing of height Hup+ Hdown, filled with water. The disk is clamped to
the shaft on its inner radius and free outside. . . 7 Figure 2.2 Companion modes n = 3, s = 0 for an annular plate . . . . 15 Figure 2.3 Working scheme for the numerical model. Continuous boxes symbolize
steps solved within CFX, while dashed boxes represent steps solved with external user Fortran subroutines (Junction Box). The mesh update consists of solving Eq. (2.55). The maximum Z-displacement step is achieved with the resolution of Eqs. (2.56-2.57) using an adapted Runge-Kutta algorithm. Convergence is based on RMS criteria of con-servative control volume fluid equation residuals. . . 17 Figure 2.4 Fluid domain mesh of the Presas et al. (2015) experimental test rig in
the (r, z) plan for the Computational Fluid Dynamics (CFD) model. Elements are cubic hexagonal, regularly spaced in the θ direction at intervals of 2◦. There are 105/n elements in this mesh, where n is the
number of nodal diameters of the studied mode. . . 18 Figure 3.1 Comparison of the disk natural frequencies for modes n = ± 2, 3, 4,
s = 0 and the Presas et al. (2015) test rig geometry; dotted data corresponds to their experimental results and lines where obtained with the analytical model detailed in this section. f = |ω|/2π. . . . 20
Figure 3.2 Disk analytical frequencies for mode n = ±3, s = 0 and a large range of disk rotating speeds. The system geometry is the same for all curves, but β0 varies from 0.2 to 5. Black curves represent co- and
counter-rotating mode frequencies, and the black dotted line shows the devia-tion from the ΩD = 0 natural frequency. This deviation is largest for
β0 = 1, while the mode split magnitude increases with β0. Hydraulic
turbines typically have ΩD/2π ≤ 10 Hz. f = |ω|/2π. . . . 21
Figure 3.3 Real (top) and imaginary (bottom) parts of the Presas et al. (2015) disk natural frequencies for modes n = 2, 3, 4, s = 0. The two real natural frequencies of a single rotating mode eventually merge when the rotation speed reaches the critical speed. According to Eq. (3.3), the corresponding critical speeds are ΩC,n=2/2π = 157 Hz, ΩC,n=3/2π =
238 Hz and ΩC,n=4/2π = 331 Hz. . . . 22
Figure 3.4 Log-log stability map for the Presas et al. (2015) experimental test rig geometry. For a given value of β0, any rotation speed corresponding to
a point on the right of the line is associated with coupled-mode flutter. The shape of the boundary is similar for any disk geometry, and given by Eq. (3.3). . . 23 Figure 3.5 Rotation speed of the disk relative to the fluid in the axial gap above
the disk at the middle radius r = (a + b)/2. Results were obtained with CFD for the Presas et al. (2015) experimental test rig geometry rotating at ΩD/2π = 4 Hz; ˆz = 0 corresponds to the rotor surface, while
ˆ
z = ˆHup corresponds to the top part of the casing. The values agree
with the theoretical expression of ΩD/F/ΩD = 1 − K and K = 0.45
(dashed line). . . 24 Figure 3.6 qc/a and qs/a as a function of the elapsed dimensionless time for mode
n = 3, s = 0 and the Presas et al. (2015) geometry. The simula-tion begins with an initial sine pulse, followed by free oscillasimula-tions of the standing disk in air. Both signals have identical frequencies and the fluid damping is negligible. Structural damping is not taken into account in the CFD analysis. Here ω/2π = 616.2 Hz matches the struc-ture natural frequency. . . 25
Figure 3.7 qc/a and qs/a as a function of the elapsed dimensionless time for mode
n = 3, s = 0 and the Presas et al. (2015) geometry. The simulation begins with an initial sine pulse, followed by free oscillations of the standing disk in water. Both signals have identical frequencies and their amplitude is damped by the dense fluid. Here ω/2π = 340.3 Hz agrees with both the analytical model and the experimental data. . . 26 Figure 3.8 qc/a and qs/a as a function of the elapsed dimensionless time for mode
n = 3, s = 0 and the Presas et al. (2015) geometry. The simulation begins with an initial sine pulse, followed by free oscillations of the rotating disk in water (ΩD/2π = 4 Hz). Both signals have close but
different frequencies, which results in a beating oscillation, character-istic of free vibration under mode split. Here |ω−− ω+|/2π = 95.9 Hz
agrees with the analytical model. . . 28 Figure E.1 qc/a as a function of the elapsed dimensionless time for mode n = 3,
s = 0 and the Presas et al. (2015) geometry. The compressibility is varied through the bulk modulus B. Water corresponds to B = 2.2 GPa. Increasing B reduces the compressibility and triggers the high frequency oscillations to appear sooner. Reducing B increases the com-pressibility and delays the high frequency oscillations. Before these ap-pear, the qc/a signals are identical, regardless of B. . . . 51
Figure E.2 Fast Fourier transform of the qc/a signals shown in Figure E.1. The
low frequency corresponds to the disk structural frequency, and does not depend on the value of the bulk modulus B. The high frequency corresponds to the parasitic oscillations, and is proportional to √B. . 52
LIST OF SYMBOLS AND ACRONYMS
1DOF 1-Degree Of Freedom NDOF N-Degrees Of Freedom
AVMI Added Virtual Mass Incremental
NAVMI Non-dimensional Added Virtual Mass Incremental CFD Computational Fluid Dynamics
FEA Finite Element Analysis FSI Fluid-Structure Interactions MMS Memory Management System RK4 Runge-Kutta method
RSI Rotor-Stator Interaction
LIST OF APPENDICES
Appendix A Matlab code: numerical resolution of the modal approach for disks . 36
Appendix B Fortran routine: input parameters . . . . 43
Appendix C Fortran routine: Runge-Kutta algorithm . . . . 46
Appendix D Fortran routine: Z-displacement . . . . 49
CHAPTER 1 INTRODUCTION
The design of modern hydroelectric turbines aims at near perfect efficiency while minimizing production costs. In this context, precise knowledge of the turbine dynamical characteristics, notably the variation of the rotor natural frequencies with rotation speed and added mass of the surrounding water, is essential to assess potential resonance and resulting amplification of vibrations. Turbine design requires fast and numerically efficient frequency identification methods taking into account these effects.
The manifold complexity of hydraulic turbines makes them subject to numerous physical effects. Resonance of structures coupled to excitation sources can lead to severe fatigue damage, that can result in the loss of hydraulic runner blades if it is not avoided (Coutu et al., 2004, 2008; Liu et al., 2016). Typically, the extra stresses due to daily start-stop cycles that turbines now undergo can initiate cracks in the blades (Huth, 2005; Trivedi and Cervantes, 2017). The intricacy of hydraulic turbine design and the numerous possible vibration sources in these machines make the runner reliability a challenging and critical criteria (Dörfler et al., 2012; Presas et al., 2019). In particular in high head Francis turbines and pump-turbines, Rotor-Stator Interaction (RSI) are an unavoidable source of excitation that needs to be predicted accurately (Dörfler et al., 2012; Walton and Tan, 2016) in order to ensure the designed geometry is suitable before its manufacturing.
Fully understanding the physics at stake is crucial in order to minimize the difficulties and delays experienced during the design stage of these machines. Critical rotation speeds of spinning structures trigger unstable regimes, such as flutter (Adams, 1987; Renshaw, 1998; Kim et al., 2000). Each structural mode can typically be stabilized by increasing the stiffness. The fluid filled spaces between rotor and casing also modify this threshold (Huang and Mote, 1995), especially with narrow gaps, like in hydraulic turbines. Additionally, the coupling between the acoustical and structural natural frequencies, the radial gap, the geometrical asymmetries, and the fluid rotation are all parameters upon which rotating structure sta-bility relies (Kang and Raman, 2006a,b). The natural frequencies of runners are affected by equally numerous parameters including their rotational velocity and the influence of sur-rounding water (Egusquiza et al., 2016). Acoustical natural frequencies can decrease natural frequencies by up to 25% if coupling occurs (Bossio et al., 2017). Such frequencies are usually higher than rotation speeds for high head turbines. This review demonstrates the complexity of both runner physics and geometry. Hence, the untangling of physical phenomenon and relevant parameters of influence on runner natural frequencies requires a simplified approach.
If efficient frequency prediction methods exist for the dynamical response of low to medium head Francis runners (Coutu et al., 2008) and of Kaplan turbines (Soltani Dehkharqani et al., 2017, 2019), this is not the case for high head turbines and in particular pump-turbine runners. Vibration modeshapes of these runners are different due to the disk-like structures of the runner crown and band, which give rise to diametrical modes, as shown in Figure 1.1(a-b). Modeshapes of eigenfrequencies typically below 450 Hz are disk-like modes (Egusquiza et al., 2016), validating the disk representation of high head turbine runners for a large range of rotation speeds. Hence, the present work studies an idealized rotating plate in dense fluid. Beyond simplifying the rotor geometry to that of a disk, we can further idealize the problem with the following assumptions:
1. The rotation speed range considered for hydraulic turbine applications is low enough to neglect centrifugal forces in the disk.
2. For the selected potential flow approach, the fluid surrounding the disk is considered inviscid and adiabatic.
3. From the rotating disk reference frame, based on work from Poncet et al. (2005), the fluid is entrained to a solid-body motion at a mean velocity equal to a fraction of that of the disk.
4. The disk modes in water are the same as those of the disk in vacuum. This was confirmed by Kwak and Kim (1991) with the Rayleigh-Ritz method.
5. We consider small amplitude deformations of the disk in order to remain in the frame of linear perturbation analysis.
These assumptions allow us to orient our literature review.
Leissa (1969) compiles analytical solutions for annular plate modes and various boundary condition sets, using previous work from Southwell (1922) and Vogel and Skinner (1965) including geometries and conditions of particular interest for the present work.
Submerged structure resonance frequencies are shifted by the surrounding water effect (Liang et al., 2007; Østby et al., 2019). Part of the fluid vibrates with the structure, adding mass to the system. The plate rotation in water additionally triggers a particular resonance mech-anism, first described by Kubota and Ohashi (1991): unlike in air, forward and backward travelling waves on the disk surface trigger the so-called mode split, as a mode can be excited with two different frequencies, as shown in Figure 1.1(c-d). However, there is little informa-tion on the physical phenomenon itself. Presas et al. (2014, 2015, 2016) along with Valentín
Figure 1.1 (a) A pump-turbine runner geometry. (b) High head Francis and pump-turbine runners have disk-like modeshapes, characterized by their number n of nodal diameters (n = 3 is presented). (c) Each disk mode is composed of a co-rotating and a counter-rotating wave, with respect to the fluid rotation relative to the disk. (d) Co- and counter-rotating wave frequencies evolve with the disk rotation speed in dense fluid: the split between the two increases, while the average value decreases.
et al. (2014) analytically and experimentally studied the natural frequencies of a submerged and confined rotating disk, as well as the influence of the rotation speed and of the axial gap length. Their experimental setup consists of a rotating disk excited with a piezoelectric patch, surrounded by air or water in a fully-rigid casing. The disk response was measured with accelerometers mounted on its surface, allowing the detection of the first structural modeshapes and associated frequencies. They showed that reducing the axial gap increases the mode split effect and decreases natural frequencies.
Various methods exist to study the vibrations of rotating structures. While considering coupled structural eigenmodes may prove close to reality, it can also be difficult to implement. Modal analysis only deals with independent modes, greatly simplifying the problem to solve. Ahn and Mote (1998) analytically studied the steady-state modal response of an excited rotating disk, and identified the modes linked to forward and backward travelling waves,
along with their associated frequencies. Renshaw et al. (1994) identified the ratio of the fluid and plate densities to be one of the main influencing parameters: mode split arises when the structure interacts with dense fluids such as water. Kwak and Kim (1991) and Amabili and Kwak (1996) analytically analyzed stationary circular plates coupled with water. They used the assumed mode approximation and worked with a potential flow. The assumed shape of the potential function must satisfy the boundary conditions; it provides spatial and temporal information on the velocity field. Then, assessing the Non-dimensional Added Virtual Mass Incremental (NAVMI) factor from the potential flow and imposed modeshape, they linked the natural frequency in vacuum to that in dense fluid. This factor represents the ratio between reference kinetic energies of the disk and fluid. Amabili et al. (1996) performed the same analysis on standing annular plates, proving its applicability to the disk considered in our work. The model developed by Presas et al. (2016) includes the relative rotation between the fluid and disk, which gives rise to mode split. It expresses the axial deformation at a characteristic radius r0, and assumes that natural frequencies can be determined only from
the disk dynamical information at r = r0. The axial gap dimensions are taken into account
through the boundary conditions. Unfortunately, all of these models still lack an exhaustive understanding of the physical nature of mode split and present a non-negligible relative error on the frequency predictions.
Analytical solutions cannot be established for hydraulic turbine complex geometries. Numer-ical FSI models using Finite Element Analysis (FEA) are powerful tools for the structural design analysis of these machines (Dompierre and Sabourin, 2010; Hübner et al., 2016). This method is applicable to natural frequency prediction of standing circular plates (Hengstler, 2013), disk-fluid-disk systems, rotor-stator systems (Specker, 2016; Weder et al., 2016, 2019), and rotating disks in fluid (Weber and Seidel, 2015). This shows that the application range of numerical FSI is extremely broad, and widely used to deal with disk frequency predic-tion. However, these fully-coupled simulations are computationally expensive (Hübner et al., 2016; Nennemann et al., 2016; Biner, 2017) and are not a convenient tool to evaluate runner natural frequencies in flow in the preliminary design stage (Weber and Seidel, 2015). They also present stability issues when the fluid-to-structure density ratio increases (Wong et al., 2013), typically when the fluid is water. Therefore, a simplified approach to model high head Francis and pump-turbines would be a powerful tool in the scope of our study. Sev-eral authors suggested using a faster approach than fully-coupled FSI, such as transient flow (Soltani Dehkharqani et al., 2017) or structural-acoustical methods (Valentín et al., 2016; Escaler and De La Torre, 2018). But because of both its simplicity and efficiency for sub-merged and confined rotating disks, the most promising alternative is a modal force approach (Nennemann et al., 2016; Presas et al., 2016; Biner, 2017), for which an arbitrary number of
modes considered separately are imposed on the disk surface. A time discretization of struc-tural equations allows the computation of the displacement according to the surrounding fluid parameters and pressure field obtained with CFD. In order to numerically capture the mode split on rotating submerged disks, our work builds on the 1-degree of freedom (1DOF) oscillator modal force CFD model developed by Monette et al. (2014) and Nennemann et al. (2016). This model was originally used to predict added stiffness and damping of runner blades in flowing water.
Here we present the development of both an analytical modal analysis and a modal force CFD approach for rotating disks in dense fluid, which accurately predict the natural frequency split and drift that are observed experimentally by Presas et al. (2016) and Weder et al. (2019). Insight into the physical origin of the mode split is also given. Both models are validated by comparison with available experimental data.
CHAPTER 2 METHODOLOGY
In this section we detail the development of the structural and fluid equations leading to the analytical modal analysis, and the modal force CFD approaches. The disk material, geometry and rotation speed, as well as the fluid and casing properties are taken into account. The radial gap is only considered in the numerical model.
The disk is a rotating annular plate of density ρD, outer radius a, inner radius b, thickness h
and angular speed ΩD in the stationary reference frame. We use the cylindrical coordinates
(r, θ, z) in the stationary reference frame, and the origin is taken at the center of the disk. The rigid casing of height Hup+ Hdown is filled with a liquid of density ρF. The gap between
the disk and the top of the casing is of length Hup, while the gap between the disk and the
bottom of the casing is of length Hdown. Figure 2.1 presents the modeled geometry.
2.1 Structural model
Let us first establish the structural equations upon which rely both analytical and numerical methods. According to linear classical plate theory, the vertical displacement w of an annular plate is given by Leissa (1969) as
D∇4w + ρDh
∂2w
∂t2 = P (r, θ, t) , (2.1)
where t is the elapsed time, P (r, θ, t) is the pressure field applied to the plate, ∇4 = ∇2∇2
with ∇2 the Laplacian operator, and
D = Eh
3
12(1 − ν2), (2.2)
is the disk flexural rigidity, where E is Young’s modulus and ν is Poisson’s ratio of the disk material, most likely stainless steel for hydraulic turbine applications. We introduce n (respectively s) as the number of nodal diameters (respectively circles) of the considered mode, and its associated modeshape in vacuum Wn. In order to determine Wn, we introduce
the parameter kn defined as
kn4 = ρDhω
2 v
Figure 2.1 Studied geometry in §2.1 and §2.2: the disk with angular speed ΩD, outer radius
a, inner radius b and thickness h is confined in a rigid casing of height Hup+ Hdown, filled
with water. The disk is clamped to the shaft on its inner radius and free outside.
which yields the information on the natural angular frequency in vacuum ωv. Leissa (1969)
provides the boundary conditions for the free-clamped annular plate:
Wn(r, θ) r=b = 0 , ∂Wn(r, θ) ∂r r=b = 0 , (2.4) Vr(r, θ) r=a = 0 , Mr(r, θ) r=a = 0 , (2.5)
where Vr is the radial Kelvin-Kirchhoff edge reaction and Mr is the bending moment:
Vr = −D " ∂ ∂r(∇ 2 Wn(r, θ)) + (1 − ν) 1 r ∂2 ∂θ∂r 1 r ∂Wn(r, θ) ∂θ !# , (2.6) Mr = −D " ∂2W n(r, θ) ∂r2 + ν 1 r ∂Wn(r, θ) ∂r + 1 r2 ∂2W n(r, θ) ∂θ2 !# . (2.7)
For annular plates, the modeshape for a single mode is expressed as
Wn(r, θ) = ψn(r) einθ, (2.8)
where
ψn(r) = AnJn(knr) + BnYn(knr) + CnIn(knr) + DnKn(knr) , (2.9)
where Jn, Yn, In, Knare the Bessel functions of first and second kinds, and the modified Bessel
functions of first and second kinds respectively, and An, Bn, Cn, Dnare coefficients determined
that the complex conjugate part is omitted in order to lighten the equations. Substituting Eqs. (2.8-2.9) in Eqs. (2.4-2.5) results in a problem of the form:
M · Xn = ai · · · di .. . . .. ... aiv · · · div · An Bn Cn Dn = 0 , (2.10)
where the matrix coefficients ai ... div are given by the developed boundary conditions.
We then have det(M) = 0 as a necessary condition for the system to be solved. Solving det(M) = 0 has an infinite number of solutions which correspond to the values of k for any number of nodal circles s. This cannot be solved analytically, and k must be evaluated numerically. After eliminating the trivial solution k = 0 and moving upward, the jth solution
that nullifies det(M) corresponds to s = j −1. Then, in order to close the system of equations, we arbitrarily choose the value of Anand then solve the system to get the values of Bn, Cn, Dn,
and hence the associated modeshape. These modeshapes naturally form an orthogonal base, and we additionally choose An to make it orthonormal.
Let us apply the Galerkin method to the disk structure. The displacement is approximated with a discrete sum:
w(r, θ, t) ≈ wN(r, θ, t) = N
X
j=1
φj(r, θ)qj(t) , (2.11)
where N is the number of modes of different nodal diameters considered, φ are test func-tions that satisfy the boundary condifunc-tions and q are the generalized coordinates. Here, the test functions are chosen to correspond to the orthonormal modeshapes: φj(r, θ) = Wj(r, θ)
defined in Eq. (2.8). Calculation shows that this choice implies ∇4W
j = kj4Wj. Hence,
substituting w with wN in Eq. (2.1) leads to
D∇4wN + ρDh
∂2w N
∂t2 − P (r, θ, t) = R , (2.12)
where R is the residual. Expanding the sum of Eq. (2.11):
N X j=1 h ρDh¨qj(t) + Dkj4qj(t) i Wj(r, θ) − P (r, θ, t) = R , (2.13)
where ¨qj = ∂2qj/∂t2. According to the Galerkin method:
Z a
r=b
Z 2π
θ=0
Let us recall that because the modes form an orthonormal base, Z a r=b Z 2π θ=0 Wi(r, θ)Wj(r, θ)r drdθ = 2π(a2− b2)δij, (2.15)
where δij is the Kronecker symbol. Therefore, multiplying Eq. (2.13) by Wi and integrating
over the annular plate surface, we obtain
2π(a2− b2) N X j=1 h ρDh ¨qj(t) + Dkj4qj(t) i δij =Pei(t) , (2.16) where e Pi(t) = Z a r=b Z 2π θ=0 P (r, θ, t)Wi(r, θ) rdrdθ . (2.17)
Let us recall that this is true for any number of considered modes N chosen arbitrarily. Observing that the equations from the obtained system are decoupled, we can then generalize for any chosen mode:
2π(a2− b2)hρ
Dh ¨qn(t) + Dkn4qn(t)
i
=Pen(t) , (2.18)
where n is the number of nodal diameters. Hence, any point on the disk can be assimilated to a 1DOF mass-spring system with vertical motion for any given mode. The vertical displacement w of this point follows
Msw(t) + K¨ sw(t) =Pen(t) , (2.19)
where Ms = 2π(a2− b2)ρDh and Ks = 2π(a2− b2)Dk4 are the modal structural mass and
stiffness respectively. It can be seen that the structural modal mass only depends on the mass and geometry of the disk, regardless of the selected mode; while the structural modal stiffness depends through k on the number of nodal diameters and circles of the mode. The disk natural angular frequency in vacuum can be assessed with Eq. (2.19):
ωv = s Ks Ms = k2 s D ρDh . (2.20)
2.2 FSI analytical model
In this section we develop an analytical model based on modal analysis, for the prediction of rotating disk natural frequencies. Let us recall that modeshapes are considered identical in air and in water. Fluid above and below the disk is taken into account, but not on the side: the radial gap between the disk and casing is considered null.
At this point, because k is known, we have an expression for any annular plate modeshape given by Eq. (2.8), and associated angular frequency in vacuum given by Eq. (2.20). As we only consider a single mode with n nodal diameters, we have N = 1 with respect to Eq. (2.11), which translates into
w(r, θ, t) = Wn(r, θ)g(t) , (2.21)
with
g(t) = eiωt, (2.22)
and ω is the actual angular frequency of the structure for the considered mode.
We consider the fluid velocity V in the surrounding fluid, the form of which is assumed to be the sum of a mean flow component V0, associated with the rotating solid body motion,
and of an oscillatory component v, associated with the transverse disk motion:
V(r, θ, z, t) = V0(r) + v(r, θ, z, t) . (2.23)
The solid body motion is described by
V0 = 0 (1 − K)rΩD 0 , (2.24)
in the cylindrical reference frame, where K is the average entrainment coefficient, the value of which will be discussed in §3.2. This coefficient verifies K = ΩF/ΩD, where ΩF is the fluid
angular speed in the stationary reference frame.
We associate the oscillatory component v to the flow potential Φ:
v = ∇Φ , (2.25)
where we assume the shape of Φ to be
Φ(r, θ, z, t) = φ(r, z)einθ˙g(t) , (2.26)
where ˙g = dg/dt, g is defined in Eq. (2.22) and φ has to be determined. By convention, n > 0 (respectively n < 0) characterizes co-rotating waves (respectively counter-rotating
waves) relative to the rotating disk. This flow obeys the Laplace equation, which implies ∇2Φ = ∂2Φ ∂r2 + 1 r ∂Φ ∂r + 1 r2 ∂2Φ ∂θ2 + ∂2Φ ∂z2 = 0 . (2.27)
Substituting Φ for its expression in Eq. (2.26) yields the equation solved by φ:
∂2φ ∂r2 + 1 r ∂φ ∂r − n2 r2φ + ∂2φ ∂z2 = 0 . (2.28)
As the fluid must remain inside the defined domain, its radial velocity is null at both the inner and outer radii:
∂φ ∂r r=b = 0 , ∂φ ∂r r=a = 0 . (2.29)
Additionally, the non-penetration boundary conditions on the top and bottom casing and disk surfaces imply
∂φ ∂z z=Hup = 0 , ∂φ ∂z z=H down = 0 , (2.30) ∂φ ∂z z=0 = Dw Dt = ∂w ∂t + V0,θ r ∂w ∂θ , (2.31)
where D/Dt is the material derivative, and V0,θis the tangential velocity in the fluid reference
frame. Then, replacing w in Eq. (2.31) by its expression given by Eqs. (2.8, 2.21) yields
∂φ ∂z z=0 = 1 + n ω V0,θ r ψn(r) . (2.32)
Substituting the tangential velocity V0,θ, given by Eq. (2.24), finally provides
∂φ ∂z z=0 = 1 + nΩD/F ω ψn(r) , (2.33)
where ΩD/F = ΩD− ΩF = (1 − K)ΩD is the disk angular speed with respect to the fluid.
Similarly to Amabili et al. (1996), we introduce the Hankel transform based on Bessel func-tions to write the potential flow as
φ(r, z) =
Z ∞
0
ξ[B(ξ)e−ξz+ C(ξ)eξz]Jn(ξr) dξ , (2.34)
where the functions B(ξ), C(ξ) are to be determined with the boundary conditions. We also introduce the non-dimensional variables ˆr = r/a, η = rξ, ˆHup = Hup/a, ˆHdown = Hdown/a,
ˆ
z = z/a and τ = t/t0, where t0 =
q
ρDh/D is a reference time. The condition on the top
casing surface in Eq. (2.32) gives
B(η)e−η ˆHup − C(η)eη ˆHup = 0 , (2.35)
Then isolating C(η) and substituting its expression in Eq. (2.33) yields
Z ∞ 0 η ηB(η)1 − e−2η ˆHup Jn(ηˆr) dη = −a3 1 + nΩD/F ω ψn(ˆr) . (2.36)
Upon applying the inverse Hankel transform we obtain
ηB(η)1 − e−2η ˆHup= −a3 1 + nΩD/F ω Z 1 b/a ˆ r ψn(ˆr)Jn(ηˆr) dˆr . (2.37)
This new integral can be evaluated numerically. Ultimately, this allows us to calculate:
η[B(η) + C(η)] = −a3 1 + nΩD/F ω 1 + e−2η ˆHup 1 − e−2η ˆHup ! Z 1 b/a ˆ r ψn(ˆr)Jn(ηˆr) dˆr . (2.38)
And replacing this expression in Eq. (2.34) gives the expression of the potential flow in the fluid volume: φup(ˆr, ˆz) = −aω 1 + nΩD/F ω Z ∞ 0 H(η)Jn(ηˆr) " e−η ˆz+ eη(ˆz−2 ˆHup) 1 − e−2η ˆHup # dη , (2.39) φdown(ˆr, ˆz) = −aω 1 + nΩD/F ω Z ∞ 0 H(η)Jn(ηˆr) " e−η ˆz+ eη(ˆz−2 ˆHdown) 1 − e−2η ˆHdown # dη , (2.40) where H(η) = Z 1 b/a ˆ r ψn(ˆr)Jn(ηˆr) dˆr . (2.41)
The perturbation fluid velocity magnitude depends on the rotation speed and frequency of the considered mode; while the velocity field shape only depends on the modeshape and geometrical properties of the domain. Evaluating Eqs. (2.39-2.40) at the disk surface ˆz = 0+
yields φup(ˆr, 0) = −aω 1 + nΩD/F ω Z ∞ 0 H(η)Jn(ηˆr)Gup(η) dη , (2.42) φdown(ˆr, 0) = −aω 1 + nΩD/F ω Z ∞ 0 H(η)Jn(ηˆr)Gdown(η) dη , (2.43)
where Gup(η) = 1 + e−2η ˆHup 1 − e−2η ˆHup , Gdown(η) = 1 + e−2η ˆHdown 1 − e−2η ˆHdown , (2.44)
so that H only depends on the mode and on the disk dimensions, and Gup, Gdownonly depend
on the disk and casing dimensions.
In order to assess the influence of the surrounding fluid on the natural frequencies of the structure, we calculate the Added Virtual Mass Incremental (AVMI) factor β, which links natural frequencies in vacuum to natural frequencies in the considered fluid (Kwak and Kim, 1991; Amabili et al., 1996):
ω ωv
= √ 1
1 + β . (2.45)
The AVMI factor is expressed as the ratio between the reference kinetic energy of the sur-rounding fluid EF to the reference kinetic energy of the disk ED. On the one hand, both the
reference kinetic energies of the above and below fluid must be considered:
EF = − 1 2ρFa 2ψ θ 1 + nΩD/F ω Z 1 b/a h φdown(ˆr, 0) + φup(ˆr, 0) i ψn(ˆr)ˆr dˆr , (2.46)
where ψθ = 2π if n = 0 and ψθ = π otherwise, as results from the integration over θ. Then
replacing the flow potentials by their known expressions Eqs. (2.42-2.43):
EF = 1 2ρFa 3ψ θ 1 + nΩD/F ω 2 × Z 1 b/a Z ∞ 0 H(η)Jn(ηˆr) h Gdown(η) + Gup(η) i dη ψn(ˆr)ˆr dˆr . (2.47)
On the other hand, the reference kinetic energy of the disk is
ED = 1 2ρDa 2ψ θh Z 1 b/a ψn(ˆr)2r dˆˆ r . (2.48)
Therefore, the AVMI factor can be expressed as
β = EF ED = ρFa 1 + nΩD/F ω 2 × R1 b/a R∞ 0 H(η)Jn(ηˆr) h Gdown(η) + Gup(η) i dη ψn(ˆr)ˆr dˆr ρDh R1 b/aψn(ˆr)2r dˆˆ r . (2.49)
expression of the AVMI factor when there is no rotation: β0 = β ΩD/F=0. (2.50) Hence, β = 1 + nΩD/F ω 2 β0, (2.51)
where β0 only depends on the fluid, casing and disk geometry and material parameters, and
the nΩD/F = n(1 − K)ΩD term yields the influence of the disk rotation. Both parts depend
on the considered mode.
Manipulating the implicit expression given by Eq. (2.45) of the natural frequency as a function of β, we can write the following explicit formulation for the modal analytical prediction of rotating and submerged disk natural angular frequencies:
ω =
q
(β0+ 1)ωv2− β0(nΩD/F)2− nβ0ΩD/F
β0+ 1
. (2.52)
Some integral terms in the developed expression of β0 need to be determined numerically.
Matlab was chosen to implement the method developed in this section. Results for various sets of geometries and modes can be obtained in only a few seconds.
2.3 FSI numerical model
In this section we develop a numerical model using results of modal analysis and CFD, for the prediction of rotating disks natural frequencies. The model can be applied to arbitrarily complex structures, taking into account the radial gap between the disk and the side walls for instance. The aim for this model is purely to predict natural frequencies using solely Ansys CFX, without the structure coupling module. Hydraulic turbine efficiency assessment is out of the scope of this study.
The oscillating disk in vacuum presents two repeated frequencies for each mode, associated with a co- and counter-rotating wave (Ahn and Mote, 1998). With respect to Eq. (2.11), we need to consider N = 2 in order to capture both waves. If the disk is rotating in dense fluid, each wave has its own angular frequency, ω+ and ω−respectively. The pair of counter-phased
modes Wn,c and Wn,s that we consider is called companion modes1 (an example is given in
Figure 2.2), where the subscripts c and s refer to the cosine and sine forms of the mode
1Ws(r, θ) = W
respectively:
Wc(r, θ) = ψn(r) cos(nθ) , (2.53)
Ws(r, θ) = ψn(r) sin(nθ) . (2.54)
Replacing the solution of the previous section, the vertical displacement is now given by
w(r, θ, t) = Wc(r, θ)qc(t) + Ws(r, θ)qs(t) , (2.55)
where qcand qs are the two unknowns to be determined by coupling CFD with a mass-spring
system, namely the two generalized coordinates of the 2DOF problem. Because the mass-spring Eq. (2.19) solved by w is linear, it is equivalent to the following system solved by qc
and qs: ¨ qc(t) + ω2qc(t) = Fc(t) Ms , where Fc(t) = Z a r=b Z 2π θ=0 P (r, θ, t)Wc(r, θ) rdrdθ , (2.56) ¨ qs(t) + ω2qs(t) = Fs(t) Ms , where Fs(t) = Z a r=b Z 2π θ=0 P (r, θ, t)Ws(r, θ) rdrdθ . (2.57)
The modal forces Fc and Fs are the projections of the pressure fields on the respective
modeshapes Wc and Ws. The angular frequency ω and the modal mass Ms are unchanged
because they only depend on the structural properties. Water effects are taken into account through the modal force. Solving these equations with CFD provides time signals for qc(t)
and qs(t), from which the angular frequencies ω+ and ω− can be deduced.
Figure 2.2 Companion modes n = 3, s = 0 for an annular plate
The CFD method developed in this section was implemented with Ansys CFX 18.2. The fluid equations are solved by Ansys CFX itself, while the disk displacement according to the imposed modeshape is implemented with custom user CEL functions and Fortran routines (see Figure 2.3). Ansys CFX performs Unsteady Reynolds-Averaged Navier–Stokes (URANS) calculations with a second order backward Euler transient scheme. We use CFX
High resolution advection scheme, which means the code tries second order where possible and reduces to first order where convergence is compromised. More details on CFX model implementation can be found in Ansys CFX User’s Manual. For a mode with n nodal diameters, only a fraction 1/n of the actual geometry is represented and periodic boundary conditions are applied consequently. The fluid domain mesh (presented in Figure 2.4) is composed of approximately 104–105 cubic hexagonal elements, depending on the studied
mode. As the fluid rotates with the disk, their are higher gradients for smaller axial gaps, hence the finer mesh above the disk. This coarse mesh is sufficient for our simulations because mode split is an inviscid fluid phenomenon, hence it is not influenced by viscous effects in the boundary layer. Mesh convergence was ensured, calculating a 1.96% frequency relative error with finer meshes (refinement factor of 2 in all three dimensions). The radial gap is considered in this model.
The model first requires input parameters. The cosine and sine form of the modeshape are established using Eq. (2.8), and normalized according to Eq. (2.15). The Bessel functions are approximated with polynomials. The modal mass and rigidity of the structure are de-termined from our structural analysis in §2.1, using the disk material properties: Ms= ρDh
and Ks = Dk4 = Msωv2. The surrounding dense fluid, typically water for hydraulic turbine
applications, is considered compressible to avoid numerical instability issues due to pressure wave propagation, as indicated in the guidelines provided by Ansys. Turbulence is repre-sented with a k- model. Y+> 30 in the mesh validates this model. The time step duration
∆t is paramount to achieve numerical stability in this case of high fluid-solid density ratio (Wong et al., 2013). Typically, for steel runners in water, ρF/ρD ≈ 0.1. Depending on the
studied geometry, the choice of ∆t may be critical to stabilize the calculation. ∆t ∼ 10−6 s in our work, and t0 =
q
ρDh/D = 8.17 · 10−2 s for the disk of the Presas et al. (2015) test
rig. Hence, the dimensionless time ∆τ = ∆t/t0 ∼ 10−5 grants enough precision so that the
fluid equations time discretization scheme needs not be of high order. Moreover, the highest mesh displacement in one time step is less than a hundred times the smallest cell size.
We then initialize the model in the rotating disk reference frame by setting the domain angular velocity to ΩD. The rotating parts in the stationary reference frame are therefore considered
stationary in the rotating disk reference frame. Casing walls are defined as counter-rotating. Centrifugal and Coriolis forces are accounted for in the fluid. At first, performing a steady state computation with w = 0 imposed allows the flow to stabilize in the rotating domain. Then enabling the mesh vertical motion and applying a sine pulse to the modal force during the first steps induces movement. We then leave the system to oscillate freely.
Figure 2.3 Working scheme for the numerical model. Continuous boxes symbolize steps solved within CFX, while dashed boxes represent steps solved with external user Fortran subroutines (Junction Box). The mesh update consists of solving Eq. (2.55). The maximum Z-displacement step is achieved with the resolution of Eqs. (2.56-2.57) using an adapted Runge-Kutta algorithm. Convergence is based on RMS criteria of conservative control volume fluid equation residuals.
simulation, as shown in Figure 2.3:
1. Pressure and velocity fields are computed in the fluid domain.
2. The pressure field is integrated on the disk, for each modeshape. The second order Eq. (2.19) is converted to a first order system of equations and then solved using the Runge-Kutta algorithm.
3. The new total mesh Z-displacement is calculated according to Eq. (2.55). The mesh is then updated.
A frequency analysis of the Z-displacement time signal provides the free oscillation frequen-cies of the system for the chosen mode.
Figure 2.4 Fluid domain mesh of the Presas et al. (2015) experimental test rig in the (r, z) plan for the CFD model. Elements are cubic hexagonal, regularly spaced in the θ direction at intervals of 2◦. There are 105/n elements in this mesh, where n is the number of nodal
CHAPTER 3 RESULTS AND DISCUSSION
3.1 Analytical model results
In this section we analyse the modal analytical model for rotating and submerged disk natural frequency prediction, given by Eq. (2.52). Table 3.1 summarizes the parameters used to model the Presas et al. (2015) experimental test rig. Table 3.2 and Figure 3.1 present frequencies of this structure’s natural frequencies for modes n = 2, 3, 4 nodal diameters and s = 0 nodal circles, assessed with Eq. (2.52). Their test rig uses a rotating disk in water, confined in a rigid casing, with small radial gap, which makes it particularly relevant with regards to our analytical model hypotheses. For a given mode, the natural frequencies of both co- and counter-rotating waves are identical when the disk is stationary. The split between ω+ and
ω−then increases with the rotation speed, while the average value slightly decreases. Overall,
Figure 3.1 illustrates that the model shows good accuracy with respect to Presas et al. (2015) experimental results.
Table 3.1 Parameter values for modeling the Presas et al. (2015) experimental test rig geom-etry. The disk is made of stainless steel; the fluid is water.
E 200·109 Pa ν 0.27
ρD 7680 kg/m3 ρF 997 kg/m3
a 0.2 m b 0.025 m
Hup 0.01 m Hdown 0.097 m
h 0.008 m K 0.45
Table 3.2 Natural frequencies of modes n = ± 2, 3, 4, s = 0 obtained with the analytical model Eq. (2.52) and the Presas et al. (2015) experiments for different disk rotation speeds and the corresponding test rig geometry, and relative error . f = |ω|/2π.
[Hz] n = 2 n = 3 n = 4 fD 0 4 8 0 4 8 0 4 8 f+,exp 127.1 120.1 117.4 321.2 317.8 309.1 642.2 619 607.9 f+,ana 117.1 113.4 109.7 312.2 307.3 302.4 626.1 620.3 614.4 + 7.9% 5.6% 6.6% 2.8% 3.3% 2.2% 2.5% 0.2% 1.1% f−,exp 127.1 126.9 132.3 321.2 328.2 330.0 642.2 630.6 633.7 f−,ana 117.1 120.7 124.4 312.2 317.1 322.0 626.1 631.9 637.7 − 7.9% 7.7% 4.9% 2.8% 4.9% 2.4% 2.5% 0.2% 0.6%
Our predictive analytical equation provides information on the two physical phenomena ap-plying to disks rotating in a dense fluid. First, let us recall that n > 0 and n < 0 respectively
Figure 3.1 Comparison of the disk natural frequencies for modes n = ± 2, 3, 4, s = 0 and the Presas et al. (2015) test rig geometry; dotted data corresponds to their experimental results and lines where obtained with the analytical model detailed in this section. f = |ω|/2π.
characterize co- and counter-rotating waves relative to the rotating disk. Therefore, the −nβ0ΩD/F term in Eq. (2.52) is responsible for the mode split phenomenon: the natural
frequency is increased by the disk rotation for counter-rotating waves, while it is decreased for co-rotating waves. The mode split magnitude is then given by
ω−− ω+ =
2nβ0ΩD/F
β0+ 1
. (3.1)
Secondly, the −β0(nΩD/F)2 term inside the square root of Eq. (2.52) is responsible for the
frequency drift: it decreases the value of the frequency regardless of the propagation direction of the wave. Both of these terms are proportional to the relative rotation speed between the disk and the fluid multiplied by the number of nodal diameters. However, the frequency drift magnitude is smaller than the mode split magnitude for typical rotation speeds of high head runners.
Figure 3.2 Disk analytical frequencies for mode n = ±3, s = 0 and a large range of disk rotating speeds. The system geometry is the same for all curves, but β0 varies from 0.2 to 5.
Black curves represent co- and counter-rotating mode frequencies, and the black dotted line shows the deviation from the ΩD = 0 natural frequency. This deviation is largest for
β0 = 1, while the mode split magnitude increases with β0. Hydraulic turbines typically have
ΩD/2π ≤ 10 Hz. f = |ω|/2π.
The influence of the considered mode and of both the disk and casing geometries on the structure natural frequencies is more difficult to interpret because of the complexity of the β0 parameter. A parametric study shows that if the axial gap is large enough (typically
Hup, Hdown ≥ 0.2 a), β0 is in the order of magnitude of the density ratio multiplied by the
aspect ratio of the disk, while it tends towards infinity if this gap becomes null:
β0 Hup,Hdown≥0.2 a ∝ ρF ρD a h, H→0lim β0 = ∞ . (3.2)
Independently, it can be determined from Eq. (2.49) that β0increases with shorter axial gaps,
and with larger and thinner disks. Additional numerical tests show that β0 decreases with
the number of nodal diameters and circles. Figure 3.2 presents the influence of β0 on the
natural frequencies predicted with our analytical model. The mode split magnitude increases with β0 towards the asymptotic value of 2nΩD/F, which is also apparent from Eq. (3.1). The
Figure 3.3 Real (top) and imaginary (bottom) parts of the Presas et al. (2015) disk natural frequencies for modes n = 2, 3, 4, s = 0. The two real natural frequencies of a single rotating mode eventually merge when the rotation speed reaches the critical speed. According to Eq. (3.3), the corresponding critical speeds are ΩC,n=2/2π = 157 Hz, ΩC,n=3/2π = 238 Hz
and ΩC,n=4/2π = 331 Hz.
frequency drift magnitude presents a maximum for β0 = 1. For typical turbine runner
dimensions, rotation speeds (ΩD/2π ≤ 10 Hz) and water density, the drift represents less
than 1% of the predicted frequency. In conclusion, the frequency drift effect is negligible in terms of hydraulic turbine applications. This is not true for the frequency split, which needs to be predicted accurately.
We now consider both positive and negative frequencies (hence n > 0). Figure 3.3 presents the evolution of these natural frequencies with the rotation speed for several modes of the Presas et al. (2015) experimental test rig geometry. The frequencies become complex values for ΩD/F above a critical value ΩC. This happens when the term under the square root of Eq.
Figure 3.4 Log-log stability map for the Presas et al. (2015) experimental test rig geometry. For a given value of β0, any rotation speed corresponding to a point on the right of the line
is associated with coupled-mode flutter. The shape of the boundary is similar for any disk geometry, and given by Eq. (3.3).
the variation of pressure in the surrounding fluid (Kornecki, 1978; Kang and Raman, 2004). The associated instability is flutter (Huang and Mote, 1995; Kim et al., 2000), representing a classical Hopf bifurcation, characterized by pairs of natural frequencies with a non-zero imaginary part for ΩD/F > ΩC (Païdoussis, 1998). The critical disk to fluid rotation speed
ΩC is given by nΩC ωv = s 1 + 1 β0 . (3.3)
Figure 3.4 shows the stability map for the non-dimensional critical speed as a function of β0.
Two asymptotic regimes emerge:
lim
β0→0
ΩC = ∞ , lim β0→∞
ΩC = ωv/n . (3.4)
This second part in Eq. (3.4) shows that critical speeds are much above the hydraulic turbine applications range for any mode and typical geometries.
3.2 Numerical model results
In this section we validate the hypothesis made on the entrainment coefficient K, and we analyze the modal numerical model results for stationary or rotating disks in air or water. All geometrical and physical parameters remain as given in Table 3.1, and the radial gap is 7 mm long. According to Poncet et al. (2005), the average entrainment coefficient of the flow between a radially delimited rotating and stationary frame satisfies
K ≈ 0.45 for 106 < Re = ΩDa2/ν < 4.5 · 106, (3.5)
where Re is the Reynolds number and ν is the kinematic viscosity of the fluid. Figure 3.5 presents the fluid rotation speed relative to the disk in the top axial gap, computed with CFD. The fluid obeys the no-slip boundary condition and rotates with the disk on its surface (ΩD/F = 0 at ˆz = 0), it is stationary on the casing surface (ΩD/F = ΩD at ˆz = ˆHup),
and it rotates at ΩD/F ≈ 0.55 ΩD for 0.1 < ˆz/ ˆHup < 0.9. This verifies that K ≈ 0.45 for the
tested geometry, and validates the hypothesis made in the analytical model development.
Figure 3.5 Rotation speed of the disk relative to the fluid in the axial gap above the disk at the middle radius r = (a + b)/2. Results were obtained with CFD for the Presas et al. (2015) experimental test rig geometry rotating at ΩD/2π = 4 Hz; ˆz = 0 corresponds to the rotor
surface, while ˆz = ˆHup corresponds to the top part of the casing. The values agree with the
theoretical expression of ΩD/F/ΩD = 1 − K and K = 0.45 (dashed line).
Figure 3.6 presents the qc and qs signals from Eqs. (2.56-2.57) for a disk in air. We matched
displace-Figure 3.6 qc/a and qs/a as a function of the elapsed dimensionless time for mode n = 3,
s = 0 and the Presas et al. (2015) geometry. The simulation begins with an initial sine pulse, followed by free oscillations of the standing disk in air. Both signals have identical frequencies and the fluid damping is negligible. Structural damping is not taken into account in the CFD analysis. Here ω/2π = 616.2 Hz matches the structure natural frequency.
ment appears as periodic and harmonic, with angular frequency ω+ = ω−. With low density
fluids such as air at 25◦C, the influence of the fluid and of the disk rotation on the natural frequencies is negligible, as Eq. (3.2) implies β0 ∼ 10−3 1 for geometries of interest. Hence,
the identical frequencies for both signals. The fluid damping is also very low.
Figure 3.7 presents the qc and qs signals for the same stationary disk, in water. With no
rotation, we still have ω+ = ω−. However, the signal amplitude now decreases with time.
Computed frequencies for several modes are compared, and agree, with the Presas et al. (2015) experimental results and with Eq. (2.52) in Table 3.3. The presence of the radial gap in the CFD model is thought to be largely responsible for the small differences between analytical and numerical results. With high density fluids such as water, the fluid influence is no longer negligible, as Eq. (3.2) implies β0 ∼ 1 for geometries of interest. For non-rotating
disks, the two main effects of the fluid are the decrease of structural natural frequencies and the damping of the displacement amplitude.
Figure 3.8 presents the qcand qssignals for the same disk, rotating in water. The signal is still
Figure 3.7 qc/a and qs/a as a function of the elapsed dimensionless time for mode n = 3,
s = 0 and the Presas et al. (2015) geometry. The simulation begins with an initial sine pulse, followed by free oscillations of the standing disk in water. Both signals have identical frequencies and their amplitude is damped by the dense fluid. Here ω/2π = 340.3 Hz agrees with both the analytical model and the experimental data.
the mode split phenomenon, because of the fluid’s different interaction with the co- and counter-rotating waves, as shown in Eq. (2.52). Both signals now show a beat envelope, due to the presence of two close frequencies: ω+ and ω−. The high frequency can be interpreted
as the natural angular frequencies average (ω−+ ω+)/2, which is usually close to the
non-rotating disk natural angular frequency in water. The beat frequency corresponds to half the mode split magnitude given by Eq. (3.1). Table 3.4 compares mode split magnitudes obtained with both analytical and numerical methods for several modes; the agreeing results show how well the physics is captured. The physical interpretation of this split is that the previously stationary mode now rotates slowly at half of the mode split angular frequency magnitude (ω−− ω+)/2. The combination of both measured frequencies (ω−+ ω+)/2 and
(ω−− ω+)/2 allows determining the actual natural angular frequencies of the rotating disk
in dense fluid: ω+ = ω −+ ω+ 2 − ω −− ω+ 2 , (3.6) ω− = ω −+ ω+ 2 + ω −− ω+ 2 . (3.7)
Table 3.3 Natural frequencies of modes n = 2, 3, 4, s = 0 obtained with the analytical model Eq. (2.52), the numerical model Eq. (2.19), and experiments from Presas et al. (2015) for the stationary disk in water, and relative error . f = |ω|/2π.
[Hz] n = 2 n = 3 n = 4 fexp 127.1 321.2 642.2 fana 117.1 312.2 626.1 ana−exp 7.9% 2.8% 2.5% fnum 130.1 340.3 641.2 num−exp 2.4% 5.9% 0.2%
Table 3.4 Split magnitude of modes n = ± 2, 3, 4, s = 0 obtained with the analytical model Eq. (3.1) and the numerical model Eqs. (2.56-2.57) for the Presas et al. (2015) rotating disk at ΩD/2π = 40 Hz in water, and relative error . f = |ω|/2π.
[Hz] n = 2 n = 3 n = 4 |f−− f+|,ana 73.2 98.1 116.6
|f−− f+|,num 73.8 95.9 115.2
ana−num 0.8% 2.2% 1.2%
After a certain simulated time that depends on geometrical and numerical parameters, inter-fering high frequency oscillations appear and prevent the observation of the beating oscilla-tion, and therefore of the split. However, they can been easily avoided without modifying the disk natural frequencies by increasing the fluid compressibility, as discussed in Appendix E.
Figure 3.8 qc/a and qs/a as a function of the elapsed dimensionless time for mode n = 3,
s = 0 and the Presas et al. (2015) geometry. The simulation begins with an initial sine pulse, followed by free oscillations of the rotating disk in water (ΩD/2π = 4 Hz). Both signals have
close but different frequencies, which results in a beating oscillation, characteristic of free vibration under mode split. Here |ω−− ω+|/2π = 95.9 Hz agrees with the analytical model.
CHAPTER 4 CONCLUSION
4.1 Summary of Works
The analytical modal approach applied to disks gives information on the potential flow in the fluid domain above and below the plate. This further results in the determination of the AVMI factor β0, characterizing the fluid effect on the structural vibrations. Eventually, we
determined an analytical expression for co- and counter-rotating wave angular frequencies:
ω =
q
(β0+ 1)ωv2− β0(nΩD/F)2− nβ0ΩD/F
β0+ 1
.
This model uses two assumptions, namely that potential flow is a good approximation and that the empirical value for the entrainment coefficient to determine the effective fluid rotation ΩD/F is correct. Accepting these assumptions, the model is truly predictive, solvable in only
a few seconds, and takes into account the geometry (including axial gaps, but not the radial gap), the fluid and structure characteristics, and the disk rotation. Both the frequency split and drift that result from the disk motion in dense fluid are well captured. The split amplitude asymptotic behavior is given by
lim
β0→∞
|ω−− ω+| = 2nΩD/F .
The modal approach applied to an arbitrary number of disk modes provides a 1DOF repre-sentation of single propagating waves. We extended this model to a 2DOF reprerepre-sentation of companion mode pairs, namely co- and counter-rotating waves of single modes. The temporal unknowns qcand qs verify equations given by the Galerkin method resolution. We simulated
freely vibrating rotating disks in dense fluid using Ansys CFX fluid solver coupled to the discretized equations with a Runge-Kutta method, while imposing specific modeshapes to the structure. Fluid damping and mode split are well captured. Mode split translates into a stationary mode when observed from the reference frame rotating at (ω− − ω+)/2. Both
analytical and numerical approaches agree with experimental data from Presas et al. (2015), and provide physical interpretation of mode split.
Our work improves knowledge of the dynamical characteristics of high head hydraulic tur-bines by providing means to assess the variation of rotor natural frequencies with rotation speed and added mass of the surrounding water, which facilitates potential resonance iden-tification within a shorter time. Our parametric and stability studies additionally show that
frequency drift and flutter instability do not occur within hydraulic turbine operation range. Both models developed in this study provide fast tools for preliminary studies on high head hydraulic turbine vibrations, and ways to explore parameters influencing mode split.
4.2 Limitations
Although extremely fast, the modal analytical approach limits the frequency prediction to single independent modes for disk geometries without radial gaps. This may not be realis-tic for some vibrating systems, and restrains its use for direct hydraulic turbine frequency prediction. However, it remains an efficient tool for understanding the physics of the mode split.
The analytical CFD model provides a fast natural frequency prediction model with limited predictive value. It proves efficient for exploring the physics of hydraulic turbine related resonance phenomena, and the influence of involved parameters. The observed parasitic high frequency vibrations that may arise limit the mode split observation at low rotation speeds.
4.3 Future Research
Although we studied the rotating disk alone, Valentín et al. (2015) showed that the casing flexibility may induce coupling with the rotating disk, resulting in the modification of its natural frequencies, especially if the gaps are small. Typically, in hydraulic turbines, this issue arises when the top part of the runner vibrates with the top casing surface (Weder et al., 2019). This makes rotor-stator coupling especially relevant for future work on the matter; possibly by adapting our numerical model. Eliminating the high frequency oscillations on the latter could be achieved by transferring the structural solver from the Fortran routines to ANSYS APDL, through System Coupling; allowing the solver to perform convergence checks on both fluid and structure equations.
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